Robust Optimization Approach Research Articles

Hierarchy relaxations for robust equilibrium constrained polynomial problems and applications to electric vehicle charging scheduling

In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.

Antibiotic dose optimisation in the critically ill: targets, evidence and future strategies.

To highlight the recent evidence for antibiotic pharmacokinetics and pharmacodynamics (PK/PD) in enhancing patient outcomes in sepsis and septic shock. We also summarise the limitations of available data and describe future directions for research to support translation of antibiotic dose optimisation to the clinical setting. Sepsis and septic shock are associated with poor outcomes and require antibiotic dose optimisation, mostly due to significantly altered pharmacokinetics. Many studies, including some randomised controlled trials have been conducted to measure the clinical outcome effects of antibiotic dose optimisation interventions including use of therapeutic drug monitoring. Current data support antibiotic dose optimisation for the critically ill. Further investigation is required to evolve more timely and robust precision antibiotic dose optimisation approaches, and to clearly quantify whether any clinical and health-economic benefits support expanded use of this treatment intervention. Antibiotic dose optimisation appears to improve outcomes in critically ill patients with sepsis and septic shock, however further research is required to quantify the level of benefit and develop a stronger knowledge of the role of new technologies to facilitate optimised dosing.

A distributionally robust optimization approach for the potassium fertilizer product transportation considering transshipment through crossdocks

Potassium fertilizer is an essential input for agricultural productivity, and plays a critical role in various plant processes, influencing water uptake, enzyme activation, and photosynthesis. The efficient delivery of potassium fertilizer products to the end-users, typically farmers and agricultural enterprises, is of utmost importance. Due to the long traveling distance between the supplier and customers, decision makers of supplier normally set up multiple crossdocks to aid in transition and storage of potassium fertilizer products. In this situation, making an optimal transportation plan as well as the inventory level of crossdocks will directly enhance the efficiency and effectiveness of long-distance transportation. In this study, we model this problem as a dynamic transportation problem with transshipment considering the uncertain time-varying customers’ demand. To characterize the demand information, we first construct an ambiguity set of customers’ demand based in limited historical data and then develop a distributionally robust optimization-based (DRO) framework to optimize the transportation plan and related inventory level of crossdocks simultaneously. We also propose a general approach to overcome the computational challenges of DRO by transforming the original DRO into a second-order cone programming based on duality theory. Additionally, we introduce linear decision rules to adjust the optimization strategy based on the new observed demand, thus lead the model to handle the dynamic information flow in real time. In case study, all involved data are collected or derived from a real potassium fertilizer company located at Western China. The results show our model reduces the cost by 18.07% compared to stochastic model (sample average approximation), indicating a significant effectiveness of our model on the improvement of delivery efficiency and cost saving in real dynamic logistic systems. Also, we conclude the managerial insight that decision-makers should develop a comprehensive strategy, including improving communication to ensure order status updates, planning rationally to evenly distribute orders, and proactively allocating resources to meet operational demands.

Multi-period human evacuation model for passenger ships under walking speed uncertainty

An effective evacuation model can steer passengers from unsafe zones to safety in a time-sensitive manner on passenger ships. The International Maritime Organization highly recommends constructing robust evacuation models rooted in understanding human behavior and ship design. This study addresses the challenge by focusing on walking speed, a key determinant of human behavior, the capacity of transitional points (e.g., exit stairs), and the distance of passengers to the exit stairs for a safe and rapid evacuation. This study presents a multi-period human evacuation model aimed at optimizing the evacuation time of the slowest passenger, identifying the optimal number of stairs, and planning route evacuation under passenger walking speed uncertainty affected by the heeling angle for a deck of a passenger ship.A robust optimization approach is also employed to manage uncertainty by defining uncertainty sets for passengers’ walking speeds. The computation of the price of robustness spans various test problems at various levels of conservatism. This research offers multiple managerial implications for maritime industry leaders, paving the way for more efficient evacuation planning.

Data-driven aerodynamic shape design with distributionally robust optimization approaches

We formulate and solve data-driven aerodynamic shape design problems with distributionally robust optimization (DRO) approaches. DRO aims to minimize the worst-case expected performance in a set of distributions that is informed by observed data with uncertainties. Building on the findings of the work Gotoh, et al. (2018), we study the connections between a class of DRO and robust design optimization, which is classically based on the mean–variance (standard deviation) optimization formulation pioneered by Taguchi. Our results provide a new perspective to the understanding and formulation of robust design optimization problems. It enables data-driven and statistically principled approaches to quantify the trade-offs between robustness and performance, in contrast to the classical robust design formulation that captures uncertainty only qualitatively. Our preliminary computational experiments on aerodynamic shape optimization in transonic turbulent flow show promising design results.

Joint optimization of dynamic pricing, quality, and production quantity in the presence of online reviews: A distributionally robust optimization approach

This paper investigates the joint pricing, product quality, and production quantity decisions of a firm in a two-stage scenario, considering potential market demand uncertainty and the existence of online reviews. Two decision models are established based on whether the firm adjusts product quality and prices according to consumer feedback from reviews: a static model (as a baseline) and a dynamic model. In the static model, the firm decides product quality, prices, and production quantity in the first stage, while in the second stage, the firm does not enhance product quality, and prices remain unchanged. In the dynamic model, in the first stage, the firm decides initial product quality and pricing, then in the second stage, adjusts product quality based on consumer feedback and sets pricing accordingly. We propose a distributionally robust optimization (DRO) method to address this problem, obtaining a closed-form solution for the expected profit function under the worst distribution, making it easy to determine the optimal decision. Through numerical simulation analysis, we find that the firm implementing dynamic decisions always make higher profits compared to those using static decisions. This highlights the superiority of dynamic decisions in adapting to market changes and improving product quality. We also analyze the impact of expected and standard deviation of potential market demand, as well as consumer private assessment weight, on business decisions and profits. Our research provides effective support for firms with limited market information, enabling them to better formulate joint pricing, quality, and production decisions.

Robust economic-resilient operational planning of distribution networks considering multi-microgrids black-start service

The increasing growth of natural disasters and cyber-physical attacks has made the development of resilience enhancement strategies for distribution networks a significant challenge for operators. Hence, in this paper, a black-start strategy based on multi-microgrids capability is developed as a two-layer preventive-corrective measure within a cost-effective energy management framework. The preventive layer includes regulating bilateral contracts between the active distribution network operator and the multi-microgrids owner pre-event. The corrective layer includes islanding microgrids during the event, calling the responsive loads, and economic-resilient operational planning of the active distribution network post-event. In this regard, a hybrid robust multi-objective optimization approach is applied to model the proposed black-start strategy as a risk-averse portfolio. The effectiveness of the developed model is tested on the modified IEEE 33-bus distribution network. Results such as an 83.38 % load recovery of the active distribution network, a 65.78 % reduction in distribution system operator costs, and a 131.28 % increase in multi-microgrid owner profit clearly illustrate the enhanced resilience of the active distribution network within an economically robust framework, which indicates the effectiveness of the proposed strategy.

Robust wind farm layout optimization

Wake interactions in wind farms cause losses in annual energy production (AEP) on the order of 10%. Wind farm designers optimize the layout of the farm to mitigate wake losses, especially in the dominant site-specific wind directions. As wind turbines and wind farms grow in scale, optimization becomes more complex. Offshore wind farms regularly comprise more than 100 wind turbines and are characterized by complex boundaries due to shipping lanes, neighboring wind farms, and other constraints.Layout optimization methods are broadly split between gradient-based and gradient-free approaches. Gradient-based approaches can converge quickly and perform well for smaller, academic problems but are often sensitive to initial conditions and tuning parameters and require expert knowledge to use. On the other hand, gradient-free approaches can be more robust to problem complexities. We present a robust layout optimization approach based on a random search algorithm. The algorithm is intended for those who are not optimization experts and has few tuning parameters that need specification to achieve satisfactory results. Unlike off-the-shelf methods, which use generally available, non-domain-specific optimization routines that accept as inputs an optimization function and constraint definitions, this approach takes advantage of the relative computational costs of the different evaluations by evaluating cheaper computations first (boundary and minimum distance constraints) and running expensive AEP evaluations only if all other checks pass. Moreover, an outer genetic algorithm allows multiple solutions to evolve in parallel, enabling rapid solution development on high-performance computers. We discuss the relative ease of selecting necessary tuning parameters and demonstrate the efficacy of the genetic random search on a complex layout problem consisting of placing 70 turbines in a nonconvex and unconnected boundary region.

A data-driven target-oriented robust optimization framework: bridging machine learning and optimization under uncertainty

ABSTRACT The target-oriented robust optimization (TORO) approach converts the original objectives to system targets and instead maximizes an uncertainty budget or robustness index. Machine learning techniques are used to tighten uncertainty sets which also reduce the pessimism of robust solutions. However, existing approaches operate only on a one-way sequential flow, where the data estimation and optimization modules implement their tasks independently. This overlooks the potential of capturing feedback within the solution framework to update forecasts and decisions based on the recent realizations of the uncertainty and system outcomes. This research proposes a novel closed-loop data-driven TORO framework, leveraging on the power of machine learning and feedback systems for optimization under uncertainty. The framework provides an array of solutions for various risk appetites and supports efficient decision-making as computational tractability is retained. A hypothetical case study is solved on a combined inventory and routing problem to demonstrate its applicability and features.

Robust optimal control for uncertain wheeled mobile robot based on reinforcement learning: ADP approach

This paper presents a robust optimal control approach for the wheel mobile robot system, which considers the effects of external disturbances, uncertainties, and wheel slipping. The proposed method utilizes an adaptive dynamic programming (ADP) technique in conjunction with a disturbance observer. Initially, the system's state space model is formulated through the utilization of kinematic and dynamic models. Subsequently, the ADP method is employed to establish an online adaptive optimal controller, which solely relies on a single neural network for the purpose of function approximation. The utilization of the disturbance observer in conjunction with the compensation controller serves to alleviate the effects of disturbances. The Lyapunov theorem establishes the stability of the complete closed-loop system and the convergence of the weights of the neural network. The proposed approach has been shown to be effective through simulation under the effect of the disturbances and the change of the desired trajectory.

Optimizing smart grid performance: A stochastic approach to renewable energy integration

Smart grids offer numerous possibilities for developing and executing sustainable and efficient electricity supply chains that exhibit increased resilience to disturbances. The exploration of the Electric Supply Chain Network (ESCND) design problem using smart grids has been limited. This article aims to tackle this challenge through the application of a robust multi-objective optimization approach, encompassing three economic goals (maximizing profit), environmental goals (minimizing greenhouse gas emissions), and resilience goals (maximizing network resilience). Additionally, the optimization process takes into account the dual objectives of maximizing efficiency and minimizing energy loss by incorporating relevant cost considerations. The proposed methodology incorporates distinctive features of smart grids, such as demandside management programs, microgrid structures, bidirectional distribution lines, and consumer–supplier nodes. It addresses various interconnected decisions, including location placement, capacity expansion, load allocation, and pricing. To solve the formulated model, a potent combination of multi-objective optimization methods based on cutting-plane algorithms and AUGMECON2 is introduced. Subsequently, the proposed model and solution approach are applied to a practical case study, and the obtained results are comprehensively examined. The findings suggest that, despite potential contradictions between economic and environmental objectives, the implementation of smart grids can concurrently improve environmental performance and network flexibility.

A Heuristic Robust Design Optimization Approach for Batch Production Permanent Magnet Actuator Consider Various Uncertainties and Life-Cycle Quality Loss

A Heuristic Robust Design Optimization Approach for Batch Production Permanent Magnet Actuator Consider Various Uncertainties and Life-Cycle Quality Loss

Bi-level distributionally robust optimization model for low-carbon planning of integrated electricity and heat systems

Interdependence among diverse energy forms within integrated electricity and heat systems (IEHSs) holds the potential to harness cooperation effects. This paper proposes a bi-level optimization model for the low-carbon planning of IEHSs, which realizes the coordination between IEHSs and their upstream networks. The precise carbon emissions of IEHSs are traced through carbon emission flows of components, power distribution networks, and district heating networks. Moreover, the distributionally robust optimization approach is employed to account for the inherent uncertainty in renewable power outputs faced by IEHS planning, by incorporating the moment information into an ambiguity set. The bi-level distributionally robust planning model of IEHSs is initially reconfigured into a second-order cone structure, employing the strong duality theory, second-order cone duality theory, and linear decision rules. Subsequently, an iterative approach is introduced to attain convergence of the reformulated bi-level model. Numerical results demonstrate the effectiveness and superiority of the proposed approach for reducing carbon emissions and economic costs of IEHSs.

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

Uncertainty in the organic Rankine cycle (ORC) system and the highly complex ORC models pose challenges to optimal operation. A data-driven robust parametric optimization for the ORC system is proposed to ensure high and stable performance under uncertainty. The uncertainty of the cyclic variables is estimated by their distribution parameters, and the average (expected) thermodynamic performance (the net output power of the ORC) is maximized as the optimization objective while minimizing its variance. An artificial neural network with the rectified linear unit is used as a surrogate model for optimization. Then the robust parametric optimization problem can be transformed into a mixed-integer linear optimization problem with a chance-constrained form, and the robust optimal solution can be solved quickly. Case studies show that the solution time of the robust optimization problem using the proposed method is about 1.4 s, which is much less than obtaining the optimal solution based on ORC mechanism models. Meanwhile, the optimal operating condition derived by the proposed robust optimization approach outperforms that obtained by the traditional deterministic strategy. The proposed approach not only improves the robustness of the system but also demonstrates the importance of considering uncertainty in parametric optimization.

Wasserstein distributionally robust optimization for train operation and freight assignment in a metro-based underground logistics system

As a shared passenger-and-freight transportation mode, the metro-based underground logistics system (M-ULS) can significantly alleviate negative effects caused by ground transportation vehicles and improve urban sustainability. Against this background, this paper addresses dedicated freight train operation and freight assignment problems in the M-ULS without affecting passenger service during the off-peak period. More importantly, this paper proposes a flexible skip-stop strategy to reduce the increased operational cost of freight trains. To hedge against uncertainty, this paper formulates a distributionally robust chance-constrained programming model by limiting uncertain freight demands to a Wasserstein ambiguity set. Moreover, the distributionally robust optimization model for a Wasserstein ball with the 1-norm can be transformed into a mixed-integer linear program, which allows for an efficient solution via CPLEX, thereby enabling metro operators to use it. To implement this study in real-life applications, a case study is conducted based on the Beijing Subway Yizhuang Line. The computational results verify the superiority of the skip-stop strategy and distributionally robust optimization approach proposed in this study. Finally, some managerial suggestions are provided for the metro operators.

Modeling a robust 2-service community healthcare network design problem with central hospital congestion

We address a more realistic variant of the community healthcare network design problem, considering a 2-service network framework and hospital congestion structures. Our objective is to minimise the total expected cost through strategic location of central hospitals and allocation of community health stations. Notably, each community health station is allocated to exactly two central hospitals. We also account for uncertainties in resident medical demands, which are challenging to predict accurately due to their volatility. To tackle this, we propose an adaptive distributionally robust optimisation approach that confines uncertain variables to an ambiguity set reflecting demand distribution characteristics. To enhance tractability, we reformulate the distributionally robust model as a mixed-integer linear programme within the specified ambiguity set. Numerical experiments on a real case demonstrate the validity and superiority of the proposed model. Finally, the impact of uncertainty is discussed in depth and some managerial insights for healthcare managers are summarised.

Solving the waste bin location problem with uncertain waste generation rate: A bi-objective robust optimization approach.

An efficient municipal solid waste (MSW) system is critical to modern cities in order to enhance sustainability and liveability of urban life. With this aim, the planning phase of the MSW system should be carefully addressed by decision makers. However, planning success is dependent on many sources of uncertainty that can affect key parameters of the system, for example, the waste generation rate in an urban area. With this in mind, this article contributes with a robust optimization model to design the network of collection points (i.e. location and storage capacity), which are the first points of contact with the MSW system. A central feature of the model is a bi-objective function that aims at simultaneously minimizing the network costs of collection points and the required collection frequency to gather the accumulated waste (as a proxy of the collection cost). The value of the model is demonstrated by comparing its solutions with those obtained from its deterministic counterpart over a set of realistic instances considering different scenarios defined by different waste generation rates. The results show that the robust model finds competitive solutions in almost all cases investigated. An additional benefit of the model is that it allows the user to explore trade-offs between the two objectives.

A risk-averse distributionally robust optimisation approach for drone-supported relief facility location problem

Drone-supported last-mile humanitarian logistics applications play a crucial role in the rapid and efficient delivery of essential relief items, such as medicine, blood, and vaccines, during disaster and emergency situations. This paper explores a novel drone-supported relief facility location problem (DSRFLP) aimed at establishing an effective drone-supported last-mile humanitarian logistics system. The problem involves making joint decisions for both pre-disaster and post-disaster phases while considering the characteristics of drone-based delivery operations and uncertain demands. In the pre-disaster phase, we make the decisions regarding the locations of drone-supported relief facilities, inventory prepositioning of relief items, assignment of drones to the opened facilities, and allocation of drones to disaster demand sites. In the post-disaster phase, we make decisions related to delivery quantities. To tackle the challenge of incomplete demand distribution information in chaotic disaster environments, we establish a distributionally robust optimisation (DRO) model to handle the uncertainty of demands. This model adopts worst-case mean Conditional Value-at-Risk as the risk measurement, reflecting the risk-averse attitude of humanitarian organisers. In this paper, three ambiguity sets (box, ellipsoidal, and polyhedral) are considered to describe the ambiguity distributions of demands. To overcome the computational challenge, we reformulate the DRO model under three ambiguity sets into two mixed-integer linear programming models and one second-order cone programming model, which can be efficiently solved by off-the-shelf solvers. Furthermore, we validate our proposed DRO model through a small-scale example and a large-scale case study based on the Lushan earthquake in China. The computational outcomes underscore the superior performance of the proposed DRO model to mitigate the impact arising from incomplete probability distributions. We propose managerial implications and insights to support the decision-making of humanitarian organisations based on the experimental results. Finally, we propose two extended models to incorporate multiple relief items and equity constraints in priority settings and conduct numerical experiments to adapt to various real-world disaster scenarios.

Data-driven Wasserstein distributionally robust dual-sourcing inventory model under uncertain demand

Dual-sourcing inventory management, which is aimed at replenishing inventory through two supply sources, has been extensively incorporated across various industries as it can mitigate supply chain related operational risks. Given the practical relevance of this framework, many dual-sourcing inventory models based on stochastic and robust optimization approaches have been developed. However, these approaches encounter challenges such as the curse of dimensionality or solution conservativeness. In this study, we developed a data-driven distributionally robust optimization model for dual-sourcing inventory management under uncertain demand conditions, in which partial information regarding the distribution of the uncertain demand is available. A tractable model was constructed to solve the problem, and an optimal solution was derived in a closed-form expression. Numerical experiments were conducted to evaluate the performance of the proposed model in comparison with benchmark models in terms of the order-, stock-, and rolling-horizon-related parameters and demand distributions. The results demonstrated the benefit of adopting the dual-sourcing strategy in inventory management based on the distributionally robust optimization approach. In addition, the proposed model outperformed the benchmark models in terms of mitigating the bullwhip effect.

A data-driven distributionally robust optimization approach for the core acquisition problem

Reusing electric vehicles (EV) batteries that reach the end of their useful first life is an environmental and cost-competitive option; however, the process of recycling EV batteries is not yet mature. Due to complex electrochemical reactions and physical conditions, the quality of used EV batteries (cores) is highly uncertain. The remanufacturer needs to make the acquisition decision under quality distributional ambiguity. Perfect quality distribution of cores cannot be known to the remanufacturer in practice. We develop distributionally robust optimization models based on phi-divergence measures and the imprecise Dirichlet model (DRO-IDM) to derive robust decisions. First, we find that the bounds of quality probability intervals are identified solely based on the collected data by introducing the imprecise Dirichlet model. The derived finite-sample boundary can reduce the scope of the uncertainty set and avoid the no-direction search issue. Second, our models can hedge against distributional uncertainty, reduce the probability of a robust solution that deviates from the optimal solution, and correct bias in decision making. Third, we extend the DRO-IDM to develop data-driven models, that can reassess the value of multisource quality information to improve the estimation accuracy of core quality and maximize the remanufacturer’s profit. Our study provides new insights for remanufacturers: the new remanufacturing process proposed in our work can assist remanufacturers in utilizing the values of cores without disassembly; the information-aware algorithm can offer the remanufacturing sector a valuable tool for efficiently filtering out invalid information in optimizing acquisition decisions; this capability empowers decision-makers to leverage multiple sources of information and expedite the process of digital transformation in remanufacturing; our approach can also provide a manner of integrating information fusion and distribution learning into remanufacturing.